Measurement and Comparison

Date: 04/30/99 at 09:43:41
From: Joanne Tranter
Subject: Measurement

Why is comparison the key to measurement?

Date: 04/30/99 at 18:11:19
From: Doctor Rick
Subject: Re: Measurement

Hi, Joanne.

I don't know the context of your question - why you are asking it - so 
I don't know what kind of answer you need. But I agree with the 
statement.

It's easiest to see when you measure length. You measure the width of 
a sheet of paper, for instance, by putting a ruler next to it and 
COMPARING the width with the positions of the markings on the ruler. 
You determine that it is between the 8 1/2 and 8 9/16 inch markings, 
perhaps. This means that the length is greater than 8 1/2 and less 
than 8 9/16 inches - those are comparisons.

Where did the ruler come from? When it was manufactured, it (or the 
machine that made it, at least) was calibrated against a standard 
ruler. That standard can trace its lineage to a metal bar of a certain 
composition, kept in certain conditions, in Washington or London or 
Paris. 

Such a standard bar was once the official definition of the foot, or
whatever. Now the inch is defined as exactly 2.54 cm, and the meter is
defined in terms of a certain number of wavelengths of light produced 
by a certain kind of atom under certain conditions.

The "certain conditions" in both the old and new standards reflect the 
need to have a standard, against which to compare our measuring 
instruments, that does not change over time. The particular atom was 
chosen for this reason.

Here is a reference on the history of the length standard, from the US
National Institute of Standards and Technology (NIST):

  http://physics.nist.gov/cuu/Units/meter.html   

The same story can be told about weights. A typical bathroom scale 
measures your weight by the compression of a spring; it is calibrated 
by checking how much the spring is compressed by certain known 
weights, so that measurement with the scale consists of comparing the 
compression caused by your body with the compression caused by these 
known weights.

But where precision is needed, a spring scale is not used. Instead, a
balance is used - a simple two-pan balance, or a triple-beam balance. 
In either case, the  object to be weighed is actually compared against 
a standard weight that is right there on the scale. (With a triple-
beam balance, just a few weights can be compared with any test weight 
by using a lever principle; the lengths of the lever arms must have 
been calibrated by comparison with a standard when the balance was 
made.)

We have had questions about whether a balance measures mass or weight 
(the force of gravity on an object). The answer is that, though it 
depends on gravity and compares the force of gravity on the test mass 
and the standard mass, it is comparing against a standard MASS. If the 
balance were taken to the moon, the forces of gravity on the two 
masses would be different, but the COMPARISON would be exactly the 
same because the forces are reduced by exactly the same amount. 

In contrast, the bathroom scale would indicate that you weigh a lot 
less, because it is actually comparing the FORCE of gravity against 
the force exerted by the spring. It would need to be seriously 
recalibrated on the moon!

If this isn't the kind of answer you are looking for, ask again.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/